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A symmetric sequence of trigonometric orthogonal functions. (English) Zbl 1441.34042

Summary: By using the extended Sturm-Liouville theorem for symmetric functions, we introduce a new differential equation, which generalizes the well-known differential equation of Fourier trigonometric sequences, and obtain its explicit solution. We then prove that the obtained solution is orthogonal with respect to the constant weight function on \([0,\pi]\) and compute its norm square value in the sequel.

MSC:

34B24 Sturm-Liouville theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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[1] Andrews, G. E.; Askey, R.; Roy, R., Special Functions. Encyclopedia of Mathematics and its Applications, 71 (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0920.33001
[2] Arfken, G., Mathematical Methods for Physicists (1985), Academic Press Inc: Academic Press Inc New York · Zbl 0135.42304
[3] Cauchy, A. L., Sur les integrales définies prises entre des limites imaginaires, Bulletin de Ferussoc, T. III, 214-221 (1825), in; (Oeuvres de A. L. Cauchy, 2 serie, T. II (1958), Gauthier-Villars: Gauthier-Villars Paris), 59-65
[4] Jirari, A., Second-order Sturm—Liouville difference equations and orthogonal polynomials, Mem. Am. Math. Soc., 542, 138 (1995) · Zbl 0817.39004
[5] Khan, S.; Riyasat, M., Differential and integral equations for the 2-iterated Appell polynomials, J. Comput. Appl. Math., 306, 116 (2016) · Zbl 1339.33024
[6] Khan, S.; Riyasat, M.; Wani, S. A., On some classes of differential equations and associated integral equations for the Laguerre-Appell polynomials, Adv. Pure Appl. Math., 9(3), 185 (2018) · Zbl 1391.45013
[7] Koekoek, R.; Lesky, P. A.; Swarttouw, R. F., Hypergeometric Orthogonal Polynomials and their q-Analogues, Springer Monographs in Mathematics (2010), Springer: Springer Berlin · Zbl 1200.33012
[8] Masjed-Jamei, M., A generalization of classical symmetric orthogonal functions using a symmetric generalization of Sturm-Liouville problems, Integral Transforms Spec. Funct., 18 no. 11-12, 871 (2007) · Zbl 1133.34020
[9] Masjed-Jamei, M., A basic class of symmetric orthogonal polynomials using the extended Sturm—Liouville theorem for symmetric functions, J. Math. Anal. Appl., 325, 753 (2007) · Zbl 1131.33005
[10] Masjed-Jamei, M., A basic class of symmetric orthogonal functions with six free parameters, J. Comput. Appl. Math., 234, 283 (2010) · Zbl 1195.33068
[11] Masjed-Jamei, M., Biorthogonal exponential sequences with weight function exp \((ax^2\) + ibx) on the real line and an orthogonal sequence of trigonometric functions, Proc. Amer. Math. Soc., 136, 409 (2008) · Zbl 1128.05053
[12] Masjed-Jamei, M.; Dehghan, M., A generalization of Fourier trigonometric series, Comput. Math. Appl., 56, 2941 (2008) · Zbl 1165.42309
[13] Masjed-Jamei, M.; Koepf, W., On incomplete symmetric orthogonal polynomials of Jacobi type, Integral Transforms Spec. Funct., 21, 655 (2010) · Zbl 1206.33010
[14] Masjed-Jamei, M.; Koepf, W., On incomplete symmetric orthogonal polynomials of Laguerre type, Appl. Anal., 90, 769 (2011) · Zbl 1220.33011
[15] Masjed-Jamei, M.; Area, I., A symmetric generalization of Sturm-Liouville problems in discrete spaces, J. Difference. Equ. Appl., 19, 1544 (2013) · Zbl 1281.39007
[16] Masjed-Jamei, M.; Area, I., A basic class of symmetric orthogonal polynomials of a discrete variable, J. Math. Anal. Appl., 399, 291 (2013) · Zbl 1263.33006
[17] Nikiforov, A. F.; Uvarov, V. B., Special Functions of Mathematical Physics (1988), Birkhäuser: Birkhäuser Basel · Zbl 0624.33001
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